利用平行直接模擬蒙地卡羅法模擬具反應與非反應高速稀薄氣體熱流場之研究

Abstract

在現代科技的研究領域當中，稀薄氣體動力學扮演著非常重要的角色。能夠描述稀薄氣體流動行為的波茲曼方程式，基本上是非常難以解析與數值直接求解的。而直接模擬蒙地卡羅法(DSMC)是一種以粒子與統計為主的數值方法，它通過直接模擬分子碰撞動力學求解波爾茲曼方程式。經由大量的驗證工作讓它被視為求解波茲曼方程式中最被廣泛接受的方法，而增加計算速度亦已經成為了擴展其應用性最主要的因素。因此，為了能有效地減少計算時間並因應稀薄氣體流場未來的發展，藉由平行計算的直接模擬蒙地卡羅法，將是不可或缺的。 在本論文中，將探討兩種主要的平行運算架構之直接模擬蒙地卡羅法。包括：採用物件導向編程之平行化直接蒙地卡羅法程式(名為PDSC++)的化學反應驗證與應用，以及運用單圖形處理器(GPU)加速具剪切Cartesian 網格之平行化二維DSMC程式。上述兩部份分別簡述如下: 第一部份：本研究藉由驗證和應用碰撞能量總和模型來模擬具化學反應之極音速稀薄氣流。在一系列的基準試例中，包括重現在單一網格中的反應速率常數與非平衡瞬態的化學反應理論值及執行二維極音速流過圓柱和二維軸對稱極音速流過球體的相關驗證。最後，以探討真實三維阿波羅6號指令艙的氣動熱力學模擬結果以展示PDSC++的強大功能，並可提供未來太空載具研發的基礎資料庫或分析能量。 第二部份：本研究運用單圖形處理器(Graphics Processor Unit, GPU)加速具剪切Cartesian 網格平行二維直接模擬蒙地卡羅法。並利用可變時步法(variable time-step scheme, VTS)與瞬態調適次網格法(transient adaptive sub-cell method, TAS)法在不影響模擬精度下減少模擬所需時間。於論文中以二維極音速流通過斜面與圓柱等案例作為驗證並詳細探討VTS和TAS方法對計算精度和計算效率的影響。模擬結果顯示使用單個GPU相比於單核Intel的CPU處理能力，根據在模擬中使用的問題和GPU的類型的大小，其計算時間將可提升13倍至35倍；同時也顯示，結合VTS與TAS方法，將大量減少模擬計算時間約10倍。 在論文最後，總結截止目前為止的成果，並提出未來建議的可能研究方向與工作。

Rarefied gas dynamics has become an increasingly important research topic in the modern science and technology. It is generally very difficult to directly solve the Boltzmann equation that governs rarefied gas dynamics. The direct simulation Monte Carlo (DSMC) method, a particle-based and statistical-based method proposed by Bird for several decades, solves the Boltzmann equation via direct simulation of particle collision kinetics. Extensive verification and validation efforts have led to its greater acceptance for solving the Boltzmann equation, whereas the increase in computer speed has been the main factor behind its greater applicability. Thus, parallel processing of the DSMC method to reduce the computational time is necessary for an efficient application of the method in its future development. In this thesis, two major categories of parallel processing for the DSMC method are presented. The first is the implementation, validation and application of chemical reaction module for simulating hypersonic reactive flows in a parallel direct simulation Monte Carlo code, named PDSC++, using Object-Oriented programming. The second is the implementation, and validation of the DSMC method using a cut-cell Cartesian grid for treating geometrically complex objects on a single graphics processing unit (GPU), which are described briefly next. The first part of the thesis focuses on the implementation and verification of TCE (total collision energy) model in the PDSC++. Through various benchmark test cases, we have demonstrated successful implementation with excellent agreement with analytical results for typical dissociation, recombination and exchange reactions. These benchmarking test cases, which include reproduction of theoretical rate constants in a single cell, 2D hypersonic flow past a cylinder and 2D-axisymmetric hypersonic flow past a sphere, were performed to validate code implementation. Finally, detailed aerothermodynamics of the flown reentry Apollo 6 Command Module at 105 km was simulated to demonstrate the powerful capability of the PDSC++ in treating realistic hypersonic reacting flow at high altitude. In the second part of the thesis, a parallel two-dimensional direct simulation Monte Carlo (DSMC) method with a cut-cell Cartesian grid for treating geometrically complex objects using a single graphics processing unit (GPU) is proposed and validate. Transient adaptive sub-cell (TAS) and variable time-step (VTS) approaches were implemented to reduce computation time without loss of accuracy. The proposed method was validated using two benchmarks: 2D hypersonic flow of nitrogen over a ramp and 2D hypersonic flow of argon around a cylinder using various free-stream Knudsen numbers. We also detailed the influence of TAS and VTS on computational accuracy and efficiency. Our results demonstrate the efficacy of using TAS in combination with VTS in reducing computation time can be more than 10 times. Compared to the throughput of a single core Intel Xeon 5670 CPU, the proposed approach using a single Nvidia GPU enables 13-35 times of increase in computational speed, which varies according to the size of the problem and type of GPUs used in the simulation. Finally, the transition from regular reflection to Mach reflection for supersonic flow through a channel was simulated to demonstrate the efficacy of the proposed approach in reproducing flow fields in challenging problems. At the end of the thesis, major findings are summarized and recommended directions of future work are outlined.